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Age 14 to 16

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Find the area of the shaded region in terms of $a$ and $b$? |

Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP have equal areas. Prove X and Y divide the sides of PQRS in the golden ratio.

The area of a regular pentagon looks about twice as a big as the pentangle star drawn within it. Is it?

A circular plate rolls in contact with the sides of a rectangular tray. How much of its circumference comes into contact with the sides of the tray when it rolls around one circuit?